18th IEEE International Conference on Control Applications Part of 2009 IEEE Multi-conference on Systems and Control Saint Petersburg, Russia, July 8-10, 2009
State Feedback Control Synthesis for Networked Systems with Multiple Sensors and Actuators Yu-Long Wang and Guang-Hong Yang Abstract— This paper studies the problems of H∞ performance analysis and controller design for NCSs with multiple sensors and actuators, where communication channels are shared by multiple sensors and actuators. The proposed periodic time sequencing scheme for controller may introduce less conservatism than the common controller gain based method. Based on the structure decomposition of free-weighting matrix, sufficient conditions guaranteeing the H∞ performance of the considered system are presented, then a new switched controller design method is proposed to optimize H∞ performance of NCSs. The merit of the proposed H∞ performance analysis lies in its less conservatism, which is achieved by adopting periodic time sequencing scheme for controller and imposing no any constraint on the structure of free-weighting matrix. The simulation results illustrate the merits and effectiveness of the proposed switched controller design.
I. INTRODUCTION Networked control systems (NCSs) have seen their rapid development recently. The defining feature of an NCS is the sharing of communication channels. The introduction of communication network into control systems will lead to many advantages, however, the sharing of communication channels will inevitably lead to time delay, packet dropout, communication constraints, etc., which should be paid much attention in NCSs design. Many researchers have studied stability/stabilization of systems in the presence of network-induced delay [1]-[2]. For NCSs, time-delay terms are piecewise differentiable, and their derivatives are equal to 1 except at countable interrupted points. In [3], the upper bound of the delay derivative was taken into consideration, and this upper bound was even allowed to be greater than or equal to 1. By employing Lyapunov functionals with discontinuities, [4] established asymptotic and exponential stability theorems for delay impulsive systems. By employing the information of probability distribution of the time delay, [5] derived the delay-distribution-dependent criteria for the mean-square exponential stability of T-S fuzzy systems. [6] presented This work was supported in part by the Funds for Creative Research Groups of China (No. 60821063), the State Key Program of National Natural Science of China (Grant No. 60534010), National 973 Program of China (Grant No. 2009CB320604), the Funds of National Science of China (Grant No. 60674021, 60804024), the 111 Project (B08015) and the Funds of PhD program of MOE, China (Grant No. 20060145019). Yu-Long Wang is with the School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China.
[email protected] Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China. He is also with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China.
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improved conditions for the global asymptotic stability of a class of neural networks with interval time-varying delays. For other results dealing with time delay, see also [7]-[8] for details. There have also been considerable research efforts on H∞ control for NCSs [9]-[10]. As we can see, the main topic of the literature presented above is time delay and packet dropout, few papers study the problem of communication constraints (limited data rate, limited data bandwidth, quantization, limited communication channels, etc.) in NCSs. In fact, NCSs with communication constraints are also a hot topic in recent years [11]-[14]. The time sequencing of messages transmitted over a network that connects the plant and the controller is paid much attention recently [15]. Fault-tolerant control of NCSs were studied in [16], where the sensors, actuators and controller were inter-connected via various medium access control protocols which employing the so-called periodic communication sequence. In [17], in a decentralized control setup, local controllers were allowed to periodically communicate to each other over a network channel. [18] proposed an H∞ approach to a remote control problem where the communication was constrained, and the controller employed a periodic time sequencing scheme for message transmissions from multiple sensors and to multiple actuators. In NCSs, multiple sensors and actuators may be included to sample plant’s states and hold control inputs. Since the number of communication channels is limited, the controller may communicate with only one of the sensor/actuator pairs at a discrete-time instant. For NCSs with such communication constraints, packet dropout is unavoidable and it has been paid full consideration in [18] (time delay was not considered), and [19] studied H∞ performance optimization for NCSs with time delay. This paper is devoted to proposing new H∞ performance analysis and controller design conditions for NCSs with multiple sensors, actuators and time delay, and improving the results of [19] correspondingly. The controller, sensors, and actuators employ a periodic time sequencing scheme for message transmissions, which may introduce less conservatism than the common controller gain based method. Sufficient conditions guaranteeing the H∞ performance of the considered system are presented, and no any constraint is imposed on the structure of free-weighting matrix, which may introduce less conservatism than the method proposed in [19], then a new switched controller design method is proposed to optimize H∞ performance of NCSs. The merit of the proposed H∞ performance analysis lies in its less conservatism, which is achieved by adopting periodic time
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sequencing scheme for controller and imposing no any constraint on the structure of free-weighting matrix. This paper is organized as follows. Section 2 presents the model of NCSs with periodic time sequencing scheme for message transmissions. Section 3 is dedicated to the H∞ performance analysis and switched controller design for NCSs with multiple sensors and actuators. The results of numerical simulation are presented in Section 4. Conclusions are stated in Section 5. Notation. Throughout this paper, M T represents the transpose of matrix M. I and 0 represent identity matrices and zero matrices with appropriate dimensions, respectively. ∗ denotes the entries of matrices implied by symmetry. Matrices, if not explicitly stated, are assumed to have appropriate dimensions.
N for the n sensors. This specifies switching pattern s1 ∈ Fn+1 that at time k, the sensor indexed as s1 (mod(k, N) + 1) is allowed to send a message; if s1 (mod(k, N) + 1) is zero, no communication takes place. Similarly, we introduce the N for the m actuators. We assume switching pattern s2 ∈ Fm+1 that, at any given time, only one message can be transmitted by a sensor or the controller. At the instant k, suppose the i sensor and the j actuator are used. In Fig. 1, the switches S1 and S2 are N-periodic, and for every k ∈ Z+ , i = 1, · · · , n, j = 1, · · · , m, they can be specified as follows
i f s1 (mod(k, N) + 1) = i, S1,k = diag(0, · · · , 0, 1, 0, · · · , 0) {z } | i i f s1 (mod(k, N) + 1) 6= i, S1,k = diag(0, · · · , 0)
II. PRELIMINARIES AND PROBLEM STATEMENT Consider the networked control system depicted in Fig. 1. The generalized plant P is a discrete-time system and has a state-space equation of the following form: xk+1 = Axk + B1 uk + B2 ωk zk = Cxk + Duk
(1)
where xk ∈ Rn , uk ∈ Rm , zk ∈ Rs , ωk ∈ Rt are the state vector, control input vector, controlled output, and disturbance input, respectively, and ωk is assumed to belong to L2 [0, ∞). A, B1 , B2 , C, D are known constant matrices of appropriate dimensions and (A, B1 ) is controllable.
(2)
i f s2 (mod(k, N) + 1) = j, S2,k = diag(0, · · · , 0, 1, 0, · · · , 0) {z } | j i f s2 (mod(k, N) + 1) 6= j, S2,k = diag(0, · · · , 0)
(3)
uk = S2,k−q vk−q = S2,k−q KS1 ,S2 S1,k−p−q xk−p−q
(4)
where S1,k ∈ Rn∗n , S2,k ∈ Rm∗m . Remark 1. As shown in (2) and (3), if s1 (mod(k, N) + 1) 6= i or s2 (mod(k, N)+1) 6= j, we may choose S1,k = diag(0 · · · 0) or S2,k = diag(0 · · · 0), respectively, that is the sensor-tocontroller or the controller-to-actuator message (at the instant k) is failed to be transmitted to the destination. Suppose the sensor-to-controller and the controller-toactuator time delay are p and q, respectively, where p and q are positive constants. Then the real control input at the instant k is
where KS1 ,S2 is the switched controller gain and the switching pattern of controller gain is based on S2,k−q and S1,k−p−q . Define p + q = l, if there exist delay, multiple sensors and actuators, the system (1) can be described as follows Fig. 1.
xk+1 = Axk + B1 S2,k−q KS1 ,S2 S1,k−l xk−l + B2 ωk zk = Cxk + DS2,k−q KS1 ,S2 S1,k−l xk−l
NCSs with multiple sensors and actuators
It is assumed that there are multiple sensors and actuators that communicate with the controller; however, as a result of the sequential nature of the channel, only one of the sensor/actuator pairs can transmit a message at any discretetime instant. To meet this requirement efficiently, we employ a periodic communication scheme (see also [18], [19]) described as follows. Let the communication period be N ≥ n + m and fix the order of transmissions within the period N. Suppose that there are n sensors and m actuators, and every one is capable to transmit messages. We index the n sensors and m actuators from 1 to n and 1 to m, respectively. Let the index set for the sensors be Fn+1 := {0, 1, · · · , n}. Then, introduce the
(5)
Remark 2. As shown in (4), the switching modes of S1,k−p−q and S2,k−q are based on constant time delay q and l. If the delay is time-varying and there exists packet dropout, the system may calculate and transmit the control inputs to the actuators in advance for every feasible value of S2,k−qk and S1,k−pk −qk , and the actuators can choose the control inputs according to the sum of time delay and packet dropout pk + qk , the corresponding design methods are omitted in this paper. If not the switched controller, but common controller gain is adopted, we can use the method proposed in [20] to design H∞ controller for the considered system. Based on the discrete-time state equation (5), we will study the problem of H∞ performance analysis and controller
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Define △Vk = Vk+1 −Vk , then
design for NCSs with limited communication channels, multiple sensors and actuators. III. H∞ PERFORMANCE ANALYSIS AND CONTROLLER DESIGN
T △V1k = xk+1 Pxk+1 − xkT Pxk
(8)
T △V2k = xkT Qxk − xk−l Qxk−l −1 T △V3k = [ηkT Z ηk − ηk+i Z ηk+i ] i=−l
(9)
∑
This section is concerned with the problem of H∞ performance analysis for the system (5), and a new switched controller design method is also proposed. By employing a periodic time sequencing scheme for message transmissions of controller, sensors, and actuators, Theorem 1 presents the conditions under which the system (5) achieves the H∞ performance γ . Theorem 1. For given scalars l, θ1 , θ2 and θ3 , if there exist e Z, e Q, e matrices M e1 , symmetric positive definite matrices P, i e e M2 , M3 , VS1 ,S2 (i = 1, · · · , n), nonsingular matrix W (where W = [W1T W2T W3T ]), scalar γ > 0, such that the following LMIs hold for every ρi (i = 1, · · · , n) e 11 Λ e 12 Λ e 13 −θ1 B2 e1 0 M Λ e 22 Λ e 23 −θ2 B2 WCT e2 M ∗ Λ e 33 −θ3 B2 e2 e3 ∗ ∗ Λ ϒ M (6)
